A Theory of Super-Resolution from Short-Time Fourier Transform Measurements

Abstract

While spike trains are obviously not band-limited, the theory of super-resolution tells us that perfect recovery of unknown spike locations and weights from low-pass Fourier transform measurements is possible provided that the minimum spacing, \(\Delta \), between spikes is not too small. Specifically, for a measurement cutoff frequency of \(f_c\), Donoho (SIAM J Math Anal 23(5):1303–1331, 1992) showed that exact recovery is possible if the spikes (on \(\mathbb {R}\)) lie on a lattice and \(\Delta > 1/f_c\), but does not specify a corresponding recovery method. Candès and Fernandez-Granda (Commun Pure Appl Math 67(6):906–956, 2014; Inform Inference 5(3):251–303, 2016) provide a convex programming method for the recovery of periodic spike trains (i.e., spike trains on the torus \(\mathbb {T}\)), which succeeds provably if \(\Delta > 2/f_c\) and \(f_c \ge 128\) or if \(\Delta > 1.26/f_c\) and \(f_c \ge 10^3\), and does not need the spikes within the fundamental period to lie on a lattice. In this paper, we develop a theory of super-resolution from short-time Fourier transform (STFT) measurements. Specifically, we present a recovery method similar in spirit to the one in Candès and Fernandez-Granda
(2014) for pure Fourier measurements. For a STFT Gaussian window function of width \(\sigma = 1/(4f_c)\) this method succeeds provably if \(\Delta > 1/f_c\), without restrictions on \(f_c\). Our theory is based on a measure-theoretic formulation of the recovery problem, which leads to considerable generality in the sense of the results being grid-free and applying to spike trains on both \(\mathbb {R}\) and \(\mathbb {T}\). The case of spike trains on \(\mathbb {R}\) comes with significant technical challenges. For recovery of spike trains on \(\mathbb {T}\) we prove that the correct solution can be approximated—in weak-* topology—by solving a sequence of finite-dimensional convex programming problems.

Mathematics Subject Classification

Notes

Acknowledgements

The authors are indebted to H. G. Feichtinger for valuable comments, in particular, for pointing out an error in an earlier version of the manuscript, J.-P. Kahane for answering questions on results in [24], M. Lerjen for his technical support with the numerical results, and C. Chenot for inspiring discussions. We also acknowledge the detailed and insightful comments of the anonymous reviewers.

Let \(\varepsilon = \{\varepsilon _\ell \}_{\ell \in \Omega }\) be a sequence of complex unit-magnitude numbers. The goal is to construct a function \(c_0 \in L^\infty (\mathbb {R}^2)\) such that \((\mathcal {A}_g^*c_0)(t_\ell ) = \varepsilon _\ell \), for all \(\ell \in \Omega \), and \(\left| (\mathcal {A}_g^*c_0)(t)\right| < 1\) for all \(t \in \mathbb {R}\setminus T\), where \(T= \{t_\ell \}_{\ell \in \Omega }\) is the support set of \(\mu \). Inspired by [12], we take \(c_0\) to be of the form

Next, we show that one can find \(\alpha , \beta \in \ell ^\infty (\Omega )\) such that the interpolation conditions \((\mathcal {A}_g^*c_0)(t_\ell ) = \varepsilon _\ell \), for all \(\ell \in \Omega \), are satisfied and \(\left| \mathcal {A}_g^*c_0\right| \) has a local extremum at every \(t_\ell \), \(\ell \in \Omega \).

3.

Then, we verify, with \(\alpha , \beta \in \ell ^\infty (\Omega )\) chosen as in 2., that the magnitude of \(\mathcal {A}_g^*c_0\) is indeed strictly smaller than 1 outside the support set \(T= \{t_\ell \}_{\ell \in \Omega }\) of \(\mu \). This will be accomplished in two stages. First, we show that \(\left| \mathcal {A}_g^*c_0\right| \) is strictly smaller than 1 “away” from each point \(t_\ell \), \(\ell \in \Omega \), specifically, on \(\mathbb {R}\setminus \bigcup _{\ell \in \Omega } [t_\ell - \frac{1}{7f_c}, t_\ell + \frac{1}{7f_c}]\). We then complete the proof by establishing that \(\left| \mathcal {A}_g^*c_0\right| \) is strictly concave on each set \([t_\ell - \frac{1}{7f_c}, t_\ell + \frac{1}{7f_c}]\), \(\ell \in \Omega \), which, combined with the fact that \(\left| (\mathcal {A}_g^*c_0)(t_\ell )\right| = 1\), for every \(\ell \in \Omega \), implies that \(\left| \mathcal {A}_g^*c_0\right| \) is also strictly smaller than 1 on each of these sets.

The main conceptual components in our proof are due to Candès and Fernandez-Granda [12]. Although [12] considers recovery of measures on \(\mathbb {T}\) only and from pure Fourier measurements, we can still borrow technical ingredients from the proof of [12, Thm. 1.2]. However, the different nature of the measurements and, in particular, the case \(G= \mathbb {R}\), pose additional technical challenges relative to the proof of [12, Thm. 1.2]. Specifically, the sum corresponding to (75) in [12] is always finite, whereas here it can be infinite, which presents us with delicate convergence issues that need to be addressed properly. Further fundamental differences between the proof in [12] for the pure Fourier case and our proof stem from the choice of the interpolation kernel, which here is given by \(t \mapsto R(t){{\mathrm{sinc}}}(2\pi f_c t)\). Specifically, we do not have to impose a bandwidth constraint on the interpolation kernel. For pure Fourier measurements, on the other hand, the interpolation kernel has to be band-limited to \([-f_c, f_c]\) (Candès and Fernandez-Granda [12] use the square of the Fejér kernel which offers a good trade-off between localization in time and frequency). As already mentioned in the main body, this leads to a factor-of-two improvement in the minimum spacing condition for STFT measurements over pure Fourier measurements. Note, however, that STFT measurements, owing to their redundancy, provide more information than pure Fourier measurements. We finally note that our proof also borrows a number of technical results from [24].

where the step from (78) to (79)–(80) follows from \(g, g' \in C_b(\mathbb {R})\), (76), (77), and the fact that \(\left| g\right| \) and \(\left| g'\right| \) are both symmetric and non-increasing on \([\Delta , \infty )\), which is by the assumption \(\Delta > 4\sigma \). Note that we eliminated the dependence of the upper bound in (79)–(80) on \((\tau , f)\). It remains to establish that every sum in the upper bound (79)–(80) is finite. The minimum separation between pairs of points of \(T= \{t_\ell \}_{\ell \in \Omega }\) is \(\Delta \), by assumption. Consequently, since \(\left| g\right| \) and \(\left| g'\right| \) are both symmetric and non-increasing on \([\Delta , \infty )\), the sums in (79) and (80) take on their maxima when the points \(t_\ell \), \(\ell \in \Omega \), are equispaced on \(\mathbb {R}\) with spacing \(\Delta \), i.e., when

and the fact that \(\widetilde{R}\) is bounded, symmetric, and non-decreasing on \([\Delta , \infty )\) as a consequence of \(\Delta > 4\sigma \). The upper bounds in (88) and (89) are both finite as the series \(\sum _{n \ge 1} R(n\Delta )\) and \(\sum _{n \ge 1} \widetilde{R}(n\Delta )\) converge.

We have shown in Lemma 3 that for \(c_0 \in L^\infty (\mathbb {R}^2)\), the function \(\mathcal {A}_g^*c_0\) is in \(C_b(\mathbb {R})\). With \(c_0\) taken as in (75), \(\mathcal {A}_g^*c_0\) is not only in \(C_b(\mathbb {R})\), but also differentiable, as we show next. We start by noting that the functions u and v defined in (86) and (87) are differentiable on \(\mathbb {R}\), and their derivatives are given by

Next, we establish that \(\sum _{\ell \in \Omega } \left( \alpha _\ell u'(t - t_\ell ) + \beta _\ell v'(t - t_\ell )\right) \) converges uniformly on every compact set \([-r, r]\), \(r > 0\), so that we can apply [13, Thm. V.2.14] to show that the series in (85) can be differentiated term by term. For \(r > 0\), we have

Since the support set \(T= \{t_\ell \}_{\ell \in \Omega }\) is closed and uniformly discrete, by assumption, and \([-r, r]\) is compact, the set \(T\cap [-r, r]\), and thereby the index set \(\Omega _r\), contains a finite number of elements, say \(L_r \in \mathbb {N}\). We thus have the following

where \(p \in \{0, 1\}\). We defer the proof of \(\mathcal {U}_p\) and \(\mathcal {V}_p\), \(p \in \{0, 1\}\), mapping \(\ell ^\infty (\Omega )\) into \(\ell ^\infty (\Omega )\) to later. The equation system (93) can now be expressed as

We further upper-bound (96) using the same line of reasoning that led to (81). Specifically, we make use of the fact that V is non-increasing on \((0, \infty )\) and that the minimum distance between points in \(T\) is \(\Delta \). This implies that \(\sum _{m\in \Omega \setminus \{\ell \}} V(t_\ell - t_m)\) is maximized for

As \(n \ge 1\), we have \(R(n\Delta ) = \exp \!\left( -\frac{\pi n^2 \Delta ^2}{4\sigma ^2}\right) \le \exp \!\left( -\frac{\pi n \Delta ^2}{4\sigma ^2}\right) \), which when used in (97) leads to a further upper bound in terms of the following power series

where the inequality is thanks to \(\tanh (\pi x^2/2) \le \pi x^2/2 \le \pi x^2\), for all \(x > 0\). Therefore, the function \(\psi \) is also non-increasing on \((0, \infty )\). Since by assumption \(\Delta > 4\sigma \) and \(\Delta > 1/f_c\), we get

where in (101) we used the fact that for \(n \ge 1\), \(1/n^2 \le 1/n\) and \(R(n\Delta ) = \exp \!\left( -\frac{\pi n^2\Delta ^2}{4\sigma ^2}\right) \le \exp \!\left( -\frac{\pi n\Delta ^2}{4\sigma ^2}\right) \). Based on the upper bound (101) we can now conclude that

where the last inequality follows from \(\Delta > 1/f_c\), \(\Delta > 4\sigma \), and the fact that \(\rho \) is non-increasing on \((0, \infty )\). Finally, using (98), (100), (103), and (104) in (99), we obtain

For later use, we record that for the choices \(\alpha = (\mathcal {U}_0 - \mathcal {V}_0\mathcal {V}_1^{-1}\mathcal {U}_1)^{-1}\varepsilon \) and \(\beta = -\mathcal {V}_1^{-1}\mathcal {U}_1\alpha \), we have

Finally, we have \(A_R''(t) \le -22.1f_c^2\). Multiplying (119) with (116) leads to \(A_R(t)A_R''(t) \le -14.6f_c^2\). Exactly the same line of reasoning can be applied to get \(A_I(t)A_I''(t) \le -14.6f_c^2\), and therefore,

for all \(t \in \left[ 0, \frac{1}{7f_c}\right] \). Indeed, we have seen that \(u''(t) \le 0\) for all \(t \in \left[ 0, \frac{1}{7f_c}\right] \), which implies that \(u'\) is non-increasing on \(\left[ 0, \frac{1}{7f_c}\right] \). As \(u'(0) = 0\), this means that \(u'\) is non-positive on \(\left[ 0, \frac{1}{7f_c}\right] \). Therefore, \(\left| u'\right| \) is non-decreasing on \(\left[ 0, \frac{1}{7f_c}\right] \), which results in (123). The inequality in (124) follows from the fact that \(\left| v'\right| \) is decreasing on \(\left[ 0, \frac{1}{7f_c}\right] \), as we show next. We have

As the functions \(t \mapsto R''(t){{\mathrm{sinc}}}(2\pi f_c t)\) and \(t \mapsto 2\pi f_cR'(t){{\mathrm{sinc}}}'(2\pi f_c t)\) were shown to both be non-decreasing on \(\left[ 0, \frac{1}{7f_c}\right] \), we get that \(v'\) is non-increasing on \(\left[ 0, \frac{1}{7f_c}\right] \). Moreover, we have

Hence, \(v'\) is non-negative on \(\left[ 0, \frac{1}{7f_c}\right) \). This allows us to conclude that \(\left| v'\right| \) is non-increasing on \(\left[ 0, \frac{1}{7f_c}\right] \), which establishes (124). It remains to upper-bound the term in (122), which is done as follows:

and determining \(\alpha :=\{\alpha _\ell \}_{\ell = 1}^L\) and \(\beta :=\{\beta _\ell \}_{\ell = 1}^L\) such that the uniqueness conditions (41) and (42) are met. It turns out, however, that a more direct path is possible, namely by choosing a function \(c_0 \in L^\infty (\mathbb {T}\times \mathbb {Z})\) of slightly different form and then reducing to a case already treated in the proof of Theorem 10; this approach leads to a substantially shorter proof. We start by defining this function \(c_0 \in L^\infty (\mathbb {T}\times \mathbb {Z})\) as

Note that since \(g\) and \(\tau \mapsto c_0(\tau , k)\), \(k \in \{-K, \ldots , K\}\), are all 1-periodic, we can integrate over the interval \([-1/2, 1/2]\) in (128) (instead of [0, 1] as done in (18)) and in the remainder of the proof. We next derive an alternative expression for the function P. As in (54), we have